Suppose that inflation is 2.64% per year (rather than 1.85% as noted above), and there is no change in Amy's income growth and the return on her savings. What would Amy's first deposit (in nominal terms) have to be in order to achieve the same real value in her retirement savings account balance when she retires?

To adjust for inflation and ensure Amy achieves the same real value in her retirement savings account, we need to consider the impact of inflation on the future value of her savings. Since inflation is now given as 2.64% (matching the rate at which Amy plans to increase her savings), and there is no change in her income growth rate or the return on her savings, we will adjust the nominal values to ensure the real value of her retirement savings remains the same.

### Key Adjustments and Calculations:

1. **Understand the Inflation Adjustment:**
   - **Inflation Rate:** 2.64% (0.0264)
   - **Real Rate of Return:** The real rate of return accounts for the inflation and is calculated using the Fisher equation:
   
   \[
   1 + r_{\text{real}} = \frac{1 + r}{1 + i}
   \]

   where $ r $ is the nominal rate of return, and $ i $ is the inflation rate.

2. **Calculate the Real Rate of Return:**
   - Using the inflation rate $ i = 2.64\% $ and the nominal return $ r = 6.73\% $, we find the real rate of return.

3. **Recalculate the Required Initial Deposit:**
   - To maintain the same real value of $2,000,000 in today's dollars, we use the real rate of return and adjust the future value to reflect the purchasing power adjusted for inflation.
   - We solve for the new initial deposit $ D_1' $ that will ensure the retirement savings reach the inflation-adjusted target in real terms.

### Calculation Steps:

1. **Calculate the Real Rate of Return $ r_{\text{real}} $:**

\[
r_{\text{real}} = \frac{1 + 0.0673}{1 + 0.0264} - 1
\]

2. **Calculate the Future Value of a Growing Annuity with the Real Rate of Return:**

   Recompute the required initial deposit using the real rate of return instead of the nominal rate.

### Let's compute the real rate of return and the adjusted initial deposit.With an inflation rate of 2.64%, the adjusted initial deposit $ D_1' $ that Amy should make in her retirement savings account next year (Year 1) is approximately **$13,883.25**. 

### Summary:
To maintain the same real value of $2 million in her retirement account, considering an inflation rate of 2.64%, Amy's first deposit needs to be $13,883.25, and this amount should still increase annually at a rate of 2.64%.

### Follow-up Questions:
1. What would be Amy's deposit amount in her 20th year with this adjusted plan?
2. How much total interest would Amy earn by the time she retires with the adjusted initial deposit?
3. If inflation unexpectedly drops to 1.5% per year after 10 years, how does this affect the required future deposits?
4. How would Amy's retirement savings change if she makes deposits monthly instead of annually?
5. What would the initial deposit be if Amy aimed for a different real value at retirement, like $1.5 million or $2.5 million?

### Tip:
When planning for long-term goals like retirement, always consider the impact of inflation on your savings. Regularly reviewing your contributions and returns with an eye on inflation will help ensure your financial security in the future.

Learn how to calculate the initial deposit Amy needs for her retirement savings to maintain the same real value despite 2.64% inflation. Understand the Fisher equation for real rate of return and how it adjusts future values for inflation. Suitable for financial planning and economic analysis.

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